The broad goal of this project is improve services research and intervention, particularly for people who are homeless and mentally ill, through the use of latent group-trajectory analytic approaches. As an R34 Exploratory Development project, the study has two specific purposes. One is to examine the utility of latent group-trajectory analysis for improving services research for people who are homeless and mentally ill. The other is to compare two techniques for such analyses to understand how each may be best used in the context of services research. We pursue both goals by analyzing whether and how these two techniques- growth mixture models and optimal matching-improve research capacity to address several important issues for services research and intervention: at-risk identification, intervention effects, and service timing. Both techniques focus on the temporal dynamism of people's lives-the patterning, duration and timing of their homelessness-and classify people with similar trajectories. By conceptualizing homelessness as subgroups of trajectories formed by commonalities in the sequencing, timing and duration of homelessness, the trajectory approach and techniques will allow us to improve outcomes for people who are homeless and mentally ill through earlier identification, better timing of service provision and more accurate understanding of the effects of service provision. The project consists of several research studies for each of the aims built around two datasets typical of the kind available to researchers. One is an experimental design of a mental health intervention for sheltered homeless people; the other is an administrative dataset that is, effectively, an observational design. The relative and joint ability of each method to contribute to addressing each of the three aims identified will be analyzed. [unreadable] [unreadable]